Lévy Processes and Infinitely Divisible Distributions

by Ken-iti Sato

• ISBN13: 9780521553025
• ISBN10: 0521553024
• Format: Hardcover
• Copyright: 1999-11-13
• Publisher: Cambridge University Press

Summary

This book provides the reader with comprehensive basic knowledge of Levy processes.

Table of Contents

Preface	ix
Remarks on notation	xi
Basic examples	1	(30)
Definition of Levy processes	1	(6)
Characteristic functions	7	(7)
Poisson processes	14	(4)
Compound Poisson processes	18	(4)
Brownian motion	22	(6)
Exercises 1	28	(3)
Notes	30	(1)
Characterization and existence of Levy and additive processes	31	(38)
Infinitely divisible distributions and Levy processes in law	31	(6)
Representation of infinitely divisible distributions	37	(10)
Additive processes in law	47	(7)
Transition functions and the Markov property	54	(5)
Existence of Levy and additive processes	59	(7)
Exercises 2	66	(3)
Notes	68	(1)
Stable processes and their extensions	69	(50)
Selfsimilar and semi-selfsimilar processes and their exponents	69	(8)
Representations of stable and semi-stable distributions	77	(13)
Selfdecomposable and semi-selfdecomposable distributions	90	(9)
Selfsimilar and semi-selfsimilar additive processes	99	(5)
Another view of selfdecomposable distributions	104	(10)
Exercises 3	114	(5)
Notes	116	(3)
The Levy-Ito decomposition of sample functions	119	(26)
Formulation of the Levy-Ito decomposition	119	(6)
Proof of the Levy-Ito decomposition	125	(10)
Applications to sample function properties	135	(7)
Exercises 4	142	(3)
Notes	144	(1)
Distributional properties of Levy processes	145	(52)
Time dependent distributional properties	145	(3)
Supports	148	(11)
Moments	159	(9)
Levy measures with bounded supports	168	(6)
Continuity properties	174	(15)
Smoothness	189	(4)
Exercises 5	193	(4)
Notes	196	(1)
Subordination and density transformation	197	(40)
Subordination of Levy processes	197	(8)
Infinitesimal generators of Levy processes	205	(7)
Subordination of semigroups of operators	212	(5)
Density transformation of Levy process	217	(16)
Exercises 6	233	(4)
Notes	236	(1)
Recurrence and transience	237	(36)
Dichotomy of recurrence and transience	237	(8)
Laws of large numbers	245	(5)
Criteria and examples	250	(13)
The symmetric one-dimensional case	263	(7)
Exercises 7	270	(3)
Notes	272	(1)
Potential theory for Levy processes	273	(60)
The strong Markov property	273	(8)
Potential operators	281	(14)
Capacity	295	(18)
Hitting probability and regularity of a point	313	(15)
Exercises 8	328	(5)
Notes	331	(2)
Wiener-Hopf factorizations	333	(52)
Factorization identities	333	(12)
Levy processes without positive jumps	345	(6)
Short time behavior	351	(12)
Long time behavior	363	(6)
Further factorization identities	369	(13)
Exercises 9	382	(3)
Notes	383	(2)
More distributional properties	385	(42)
Infinite divisibility on the half line	385	(9)
Unimodality and strong unimodality	394	(9)
Selfdecomposable processes	403	(13)
Unimodality and multimodality in Levy processes	416	(8)
Exercises 10	424	(3)
Notes	426	(1)
Solutions to exercises	427	(24)
References and author index	451	(28)
Subject index	479